## Abstract

We study the correspondence between almost periodic difference operators and algebraic curves (spectral surfaces). An especial role plays the parametrization of the spectral curves in terms of, so-called, branching divisors. The multiplication operator by the covering map with respect to the natural basis in the Hardy space on the surface is the 2d+1-diagonal matrix; the d-root of the product of the Green functions (counting their multiplicities) with respect to all infinite points on the surface is the symbol of the shift operator. We demonstrate an application of our general construction to the particular covering, which generate almost periodic CMV matrices recently widely discussed. Then we study an important theme: covering of one spectral surface by another one and the related transformations on the set of multidiagonal operators (so-called Renormalization Equations). We prove several new results dealing with Renormalization Equations for periodic Jacobi matrices (polynomial coverings) and for the case of a rational double covering.

Original language | English |
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Title of host publication | Methods of Spectral Analysis in Mathematical Physics - Conference on Operator Theory, Analysis and Mathematical Physics, OTAMP 2006 |

Editors | Jan Janas, Pavel Kurasov, Ari Laptev, Ari Laptev, Sergei Naboko, Gunter Stolz |

Publisher | Springer International Publishing |

Pages | 347-389 |

Number of pages | 43 |

ISBN (Print) | 9783764387549 |

DOIs | |

State | Published - 2009 |

Externally published | Yes |

Event | Conference on Operator Theory, Analysis and Mathematical Physics, OTAMP 2006 - Lund, Sweden Duration: 1 Jan 2006 → … |

### Publication series

Name | Operator Theory: Advances and Applications |
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Volume | 186 |

ISSN (Print) | 0255-0156 |

ISSN (Electronic) | 2296-4878 |

### Conference

Conference | Conference on Operator Theory, Analysis and Mathematical Physics, OTAMP 2006 |
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Country/Territory | Sweden |

City | Lund |

Period | 1/01/06 → … |

### Bibliographical note

Publisher Copyright:© 2008 Birkhäuser Verlag Basel/Switzerland.

### Funding

Partially supported by the Austrian Founds FWF, project number: P20413-N18 and Marie Curie International Fellowship within the 6th European Community Framework Programme, Contract MIF1-CT-2005-006966.

Funders | Funder number |
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Austrian Founds FWF | P20413-N18 |

Sixth Framework Programme | MIF1-CT-2005-006966 |

## Keywords

- CMV matrices
- Expanding polynomials
- Hardy spaces
- Iterations
- Jacobi matrices
- Riemann surfaces
- Spectral theory